By using the CLI on Linux, can I make a running blender instance save its current scene to file?
No. Blender accepts cli options when starting that can include a python script to automate tasks but once started it no longer accepts commands from the cli. 

The autosave, if is hasn't been disabled, should save a temporary copy, even the unmodified startup file will regularly autosave a temporary copy. If your autosave file is old then it may indicate that the copy of blender is no longer running.

If you can't get back to the machine (or get someone else) to close it properly then the only option I can think of trying is x11vnc that can connect to an existing xorg desktop. See if you can download a copy to your home folder and run it without admin permissions.